. Prove that: 
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
and 
then prove 
be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
and 
Prove that
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
Prove that
of the pizza, where 
is a positive integer. Find 
![\[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]](http://latex.artofproblemsolving.com/e/b/2/eb28918359848dd07784a3c2c6a34b934743c016.png)
be 
.![\[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]](http://latex.artofproblemsolving.com/3/1/5/315341a7adf35bc04e40ee0685a011c9f7b76115.png)
without Newton Sums.
Is there an elegant way to prove this?
can be partitioned into 
pairs such that the sum of the elements in each pair is a power of 
.
is stylish because the set 
can be partitioned as 
,
,
.
to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
and 
,
and 
although its supposed to be a quick comparison.
chance of landing heads when flipped. Krithik is playing a game where he starts with 
point. Every turn, he flips the coin, and if it lands heads, he gains 
points. Find the expected number of turns until Krithik wins.
be an isosceles triangle with 
Let 
be a point on the circumcircle of 
Let 
intersect the extension of 
at 
Let 
be the midpoint of segment 
and let 
be the intersection of 
and 
Let the extension of 
intersect 
and the circumcircle of 
and 
respectively. Given that 
and 
compute the length of 
Y by
Let 
be a sharp-angled triangle of circumcenter 
and let 
be the feet of altitudes on the sides 
, 
, 
, respectively. Let 
be the circumcircle of , let 
is the circle of 
and let 
be a variable point about . Consider a circle tangent internally to at by and tangent externally the at 
. Prove that the line 
passes through a fixed point on the line 
.
be a sharp-angled triangle of circumcenter 
and let 
be the feet of altitudes on the sides 
, 
, 
, respectively. Let 
be the circumcircle of , let 
is the circle of 
and let 
be a variable point about . Consider a circle tangent internally to at by and tangent externally the at 
. Prove that the line 
passes through a fixed point on the line 
.
.
,
.
.
.
and radius 
.
.
on the line 
.
